Embedded newform 171.3.z.a.101.31

• Label: 171.3.z.a.101.31
• Level: $171$
• Weight: $3$
• Character: 171.101
• Analytic conductor: $4.659$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	171.z (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65941252056\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(38\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			101.31
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.62758 - 0.463312i) q^{2} +(2.70091 + 1.30579i) q^{3} +(2.93072 - 1.06670i) q^{4} +(2.10712 + 2.51117i) q^{5} +(7.70183 + 2.17971i) q^{6} +(-2.69558 - 4.66888i) q^{7} +(-2.03612 + 1.17556i) q^{8} +(5.58981 + 7.05366i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{9}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.62758	−	0.463312i	1.31379	−	0.231656i	0.527520	−	0.849543i	\(-0.323122\pi\)
							0.786267	+	0.617886i	\(0.212011\pi\)
\(3\)	2.70091	+	1.30579i	0.900303	+	0.435264i
							\(5\)	2.10712	+	2.51117i	0.421424	+	0.502233i	0.934428	−	0.356153i	\(-0.115912\pi\)
−0.513004	+	0.858386i	\(0.671467\pi\)
\(7\)	−2.69558	−	4.66888i	−0.385082	−	0.666982i	0.606698	−	0.794932i	\(-0.292494\pi\)
							−0.991781	+	0.127950i	\(0.959160\pi\)
\(11\)		−	11.1401i		−	1.01274i	−0.862316	−	0.506370i	\(-0.830987\pi\)
							0.862316	−	0.506370i	\(-0.169013\pi\)
\(13\)	9.72499	+	8.16023i	0.748076	+	0.627710i	0.934993	−	0.354665i	\(-0.115405\pi\)
							−0.186917	+	0.982376i	\(0.559850\pi\)
\(17\)	−15.2759	−	18.2051i	−0.898580	−	1.07089i	−0.997127	−	0.0757537i	\(-0.975864\pi\)
							0.0985463	−	0.995132i	\(-0.468581\pi\)
\(19\)	−16.8177	−	8.84110i	−0.885142	−	0.465321i
							\(23\)	13.9019	+	38.1951i	0.604430	+	1.66066i	0.742188	+	0.670192i	\(0.233788\pi\)
−0.137758	+	0.990466i	\(0.543990\pi\)
\(29\)	−16.2747	−	44.7145i	−0.561198	−	1.54188i	−0.817883	−	0.575384i	\(-0.804853\pi\)
							0.256685	−	0.966495i	\(-0.417370\pi\)
\(31\)	1.25868			0.0406025			0.0203013	−	0.999794i	\(-0.493537\pi\)
							0.0203013	+	0.999794i	\(0.493537\pi\)
\(37\)	3.67627			0.0993586			0.0496793	−	0.998765i	\(-0.484180\pi\)
							0.0496793	+	0.998765i	\(0.484180\pi\)
\(41\)	−19.2285	+	3.39050i	−0.468988	+	0.0826952i	−0.403148	−	0.915135i	\(-0.632084\pi\)
							−0.0658403	+	0.997830i	\(0.520973\pi\)
\(43\)	−6.18275	−	2.25034i	−0.143785	−	0.0523334i	0.269125	−	0.963105i	\(-0.413265\pi\)
							−0.412910	+	0.910772i	\(0.635488\pi\)
\(47\)	16.9215	+	46.4914i	0.360031	+	0.989178i	0.979018	+	0.203776i	\(0.0653214\pi\)
							−0.618986	+	0.785402i	\(0.712456\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.31	✓	228
9.5	odd	6		171.3.bf.a.158.31	yes	228
19.16	even	9		171.3.bf.a.92.31	yes	228
171.149	odd	18	inner	171.3.z.a.149.31	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.31	✓	228	1.1	even	1	trivial
171.3.z.a.149.31	yes	228	171.149	odd	18	inner
171.3.bf.a.92.31	yes	228	19.16	even	9
171.3.bf.a.158.31	yes	228	9.5	odd	6